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Math Help - Convergent sequence

  1. #1
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    Convergent sequence

    I am having a lot of trouble attempting this problem. I do not know what is the best way to go about proving it.

    Suppose sequence a_n > 0 and b_n = a_n + \frac{1}{a_n}<br />
    Assume a_n >= 1 for all n, and that  b_n converges. Prove that a_n converges.

    If it assumed that b_n converges but only  a_n > 0, it does not follow that a_n converges. Find the example.
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  2. #2
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    Quote Originally Posted by Rozaline View Post
    I am having a lot of trouble attempting this problem. I do not know what is the best way to go about proving it.

    Suppose sequence a_n > 0 and b_n = a_n + \frac{1}{a_n}<br />
    Assume a_n >= 1 for all n, and that  b_n converges. Prove that a_n converges.

    If it assumed that b_n converges but only  a_n > 0, it does not follow that a_n converges. Find the example.
    If b_n = a_n + \tfrac{1}{a_n} then a_n^2 - b_na_n + 1 = 0. Use the quadratic formula to solve that equation: a_n = \tfrac12\bigl(b_n\pm\sqrt{b_n^2-4}\bigr). That should tell you enough about a_n to find solutions for both parts of the problem.
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